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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeFeb 8th 2017

I added a remark to inhabited set that one can regard writing $A\neq\emptyset$ to mean “$A$ is inhabited” as a reference to an inequality relation on sets other than denial.

1. Nice observation!

Looking at that entry reminded me of a situation where the constructively sensible rendition of inhabitation is in fact non-emptiness. Let $f : X \to S$ be an $S$-scheme (in a classical context), which we visualize as an $S$-indexed family of schemes. Recall that the functor of points of $X$, $\underline{X} \coloneqq Hom_S(\cdot, X)$, is an object of the big Zariski topos of $S$ and that the internal language of that topos is in all interesting cases not Boolean.

Then $f$ is set-theoretically surjective, meaning that all its fibers are inhabited, if and only if from the point of view of the Zariski topos, $\underline{X}$ is not empty.

(The condition that $\underline{X}$ is inhabited from the internal point of view means something much stronger, namely that the projection $X \to S$ locally has a section, so that not only individual points of $S$ lift, but entire open parts.)

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeFeb 9th 2017

Nice example. In general, I find that the traditional constructivist antipathy towards logical negation is unnecessary. Often it is better to rephrase things “positively”, but plenty of sensible constructive notions do involve negation. For instance, the notion of “disjoint sets” involves a negation, and the inequality $x\le y$ of real numbers can be defined as $\neg (x\gt y)$.

2. I agree. (Of course, a practical reason for tending to prefer positive formulations is that those are one step closer to being geometric formulas, which in turn are nice because they behave excellently under pullback along geometric morphisms. For anyone secretly following this conservation, but not quite grasping the importance of geometricity, I recommend Steve Vicker’s notes on this topic.)

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeFeb 10th 2017

True. Of course, negative definitions can also occur in geometric theories, since $P\vdash\bot$ is a geometric sequent.